When a student multiplied the number $66$ by the repeating decimal, $$\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit integer $\underline{a} \ \underline{b}?$

Answer: E

Concepts:repeating decimalgeometric sequence

Difficulty rating: 1370

AMC 8AMC 10AMC 12AIME

Video solution:

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Written solution:

Let $N=10a+b,$ the two-digit integer formed by the digits $a$ and $b.$ Then $$1.\overline{ab}=1+\frac{N}{99}$$ while the terminating decimal the student used is $$1.ab=1+\frac{N}{100}.$$

The correct product exceeds the student's product by $0.5,$ so $$66\left(\frac{N}{99}-\frac{N}{100}\right) =\frac{N}{150}=0.5.$$ Hence $N=75.$

Thus, E is the correct answer.